Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2552 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#384 | C2 |
| hypercenter | #384 | D3 |
| hypocenter | C3 | #2552 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
452 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,19,20,18)(17,23,22,21);(2,3)(5,7)(8,10,11,15)(9,12,13,14)(17,22)(21,23);(4,5,6,7)(16,18,20,19)(17,21,22,23);(2,3)(8,9,11,13)(10,14,15,12)(16,17,20,22)(18,23,19,21)Polycyclic presentation:
2^7,3|a^2=f,b^2=eg,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10698229, #10698230, #10698236, … |
| maximal normal subgroup of | #10698229, #10698230, #10698236, … |
| maximal quotient of | #10717464, #10717471, #10718114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6912);